Winning Region Computation and Analysis for a UAV-vehicle Pursuit-evasion Game Based on Smooth Approximation
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摘要: 针对无人机攻击地面车辆场景下具有非零攻击半径、任务时间受限以及目标受地面约束的追逃博弈问题, 建立空地异构参与者的相对运动学模型, 并将无人机在给定时间内进入攻击范围的问题表述为有限时域获胜区求解问题. 在此基础上, 构造相应的哈密顿–雅可比–伊萨克斯变分不等式模型, 给出原始博弈的最优控制与闭式哈密顿量表达, 并针对其中由控制切换与分段结构引起的非光滑性构造平滑近似哈密顿量. 结合水平集框架、局部Lax-Friedrichs耗散和Runge-Kutta时间推进, 对全四维状态空间中的值函数进行回溯求解, 并基于固定切片、参数–相对航向角热力图及边界轮廓对速度比、角速度比和攻击半径对获胜区的影响进行分析. 数值结果表明, 在平滑近似模型与参数扫描范围下, 三类参数的增大均有助于扩大无人机获胜区, 但作用方式存在明显差异: 速度比主要决定获胜区的整体扩张, 角速度比主要表现为对局部边界形状的修正, 攻击半径则主要体现为终端命中条件放宽所带来的几何外扩. 相关结果可为无人机攻击约束设计、机动能力配置及对抗效能评估提供参考.Abstract: This paper studies a pursuit-evasion game problem involving an attacking UAV and a ground vehicle, where the UAV has a nonzero attack radius and a limited mission time, while the target is constrained to move on the ground. A relative kinematic model for the heterogeneous aerial and ground players is established, and the problem of whether the UAV can enter the attack range within a prescribed time is recast as that of solving for the finite-horizon winning region. On this basis, the corresponding Hamilton-Jacobi-Isaacs variational inequality model is constructed, and the optimal controls of the original game and a closed-form expression for its Hamiltonian are derived. To address the nonsmoothness caused by control switching and piecewise structures, a smooth-approximation Hamiltonian is further constructed. The value function is solved backward in the full four-dimensional state space using a level-set framework, local Lax-Friedrichs dissipation, and Runge-Kutta time marching. Fixed slices, parameter-relative heading angle heatmaps, and boundary contours are then used to analyze the effects of the speed ratio, angular-speed ratio, and attack radius on the winning region. Numerical results show that, under the smooth-approximation model and within the parameter ranges scanned, an increase in each of the three parameters helps enlarge the UAV winning region, but they do so in significantly different ways: The speed ratio mainly determines the overall expansion of the winning region, the angular-speed ratio mainly modifies the local boundary shape, and the attack radius mainly causes geometric outward expansion by relaxing the terminal hit condition. Related results provide references for UAV attack-constraint design, maneuverability allocation, and evaluation of engagement effectiveness.
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Key words:
- pursuit-evasion game /
- winning region /
- smooth approximation /
- parameter effect analysis
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图 1 无人机攻击地面车辆追逃场景的惯性坐标系与运动变量示意图((a)空间态势示意图; (b)水平投影视图; (c)侧视图)
Fig. 1 Illustration of the inertial coordinate system and motion variables in the UAV-ground vehicle pursuit-evasion scenario ((a) Illustration of the spatial engagement configuration; (b) Horizontal projection view; (c) Side view)
图 2 原始非光滑哈密顿量与平滑近似哈密顿量所得零水平集边界对比
Fig. 2 Comparison of zero level set boundaries obtained from the original nonsmooth Hamiltonian and the smooth-approximation Hamiltonian
图 3 基准参数下固定$ z=1 $、$ \alpha=\pi/4 $时$ x $-$ y $平面获胜区零水平集随回溯时间$ \tau $的演化
Fig. 3 Evolution of the winning-region zero level set on the $ x $-$ y $ plane with respect to backward time $ \tau $ at fixed $ z=1 $ and $ \alpha=\pi/4 $ under the baseline parameters
图 4 获胜区比例随速度比的变化
Fig. 4 Variation of the winning region proportion with the speed ratio
图 5 终端获胜区比例随速度比与相对航向角$ \alpha $的联合变化热力图
Fig. 5 Heatmap of the joint variation of the terminal winning region proportion with the speed ratio and the relative heading angle $ \alpha $
图 6 固定$ \tau=3 $、$ z=1 $、$ \alpha=\pi/4 $时, 不同速度比对应的$ x $-$ y $平面零水平集轮廓
Fig. 6 Zero level set contours on the $ x $-$ y $ plane for different speed ratios with fixed $ \tau=3 $, $ z=1 $, and $ \alpha=\pi/4 $
图 7 终端总体获胜区比例随角速度比的变化
Fig. 7 Variation of the overall terminal winning region proportion with the angular speed ratio
图 8 终端获胜区比例随角速度比与相对航向角$ \alpha $的联合变化热力图
Fig. 8 Heatmap of the joint variation in the terminal winning region proportion with the angular speed ratio and the relative heading angle $ \alpha $
图 9 固定$ \tau=3 $、$ z=1 $、$ \alpha=\pi/4 $时, 不同角速度比对应的$ x $-$ y $平面零水平集轮廓
Fig. 9 Zero level set contours on the $ x $-$ y $ plane for different angular speed ratios with fixed $ \tau=3 $, $ z=1 $, and $ \alpha=\pi/4 $
图 10 终端总体获胜区比例随攻击半径$ l $的变化
Fig. 10 Variation of the overall terminal winning region proportion with the attack radius $ l $
图 11 终端获胜区比例随攻击半径与相对航向角$ \alpha $的联合变化热力图
Fig. 11 Heatmap of the joint variation in the terminal winning region proportion with the attack radius and the relative heading angle $ \alpha $
图 12 固定$ \tau=3 $、$ z=1 $、$ \alpha=\pi/4 $时, 不同攻击半径对应的$ x $-$ y $平面零水平集轮廓
Fig. 12 Zero level set contours on the $ x $-$ y $ plane for different attack radii with fixed $ \tau=3 $, $ z=1 $, and $ \alpha=\pi/4 $
表 1 原始模型与平滑模型获胜区差异指标
Table 1 Difference metrics between the winning regions of the original and smooth models
指标 数值 四维获胜区比例(原始模型) $0.526\,\;581$ 四维获胜区比例(平滑模型) $0.482\,\;500$ 四维获胜区比例差 $0.044\,\;081$ 四维交并比 $0.833\,\;958$ 固定切片交并比 $0.843\,\;700$ 二维零水平集平均对称距离 $0.723\,\;710$ 
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表 2 代表性阈值下的敏感层驻留统计
Table 2 Residence statistics of sensitive layers under representative thresholds
统计对象 均值 中位数 $95\%$分位 边界敏感层驻留比例 $4.08\%$ $3.00\%$ $10.10\%$ 归一化切换层驻留比例 $66.89\%$ $72.75\%$ $100.00\%$ 交集敏感层驻留比例 $2.75\%$ $2.40\%$ $8.30\%$ 交集进入次数 $1.41$ $1.00$ $4.00$
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